Topology optimization has been identified as a powerful tool to improve aircraft structures for many years. Yet, innovative layouts have not been successfully implemented in commercial aircraft for several reasons. One reason identified by our research group is the lack of design constraints during topology optimization, such as buckling stability, which yields complex solutions that are not easily manufacturable. Second, the complexity of the resulting layouts makes integration with other systems highly challenging. With respect to these challenges, we propose a new heuristic layout optimization process: complexity-driven layout exploration for aircraft structures (CD-LEAS). The new process addresses the challenges of complexity and nonlinear constraints, such as buckling, in aircraft structure layout optimization. The novelty of CD-LEAS comes from the integration of a relative complexity metric as a driver to navigate the design space efficiently. Two case studies of commonly used stiffened panels are carried out to showcase the performance of the process. The results show that using complexity to navigate an explicit design space allows our process to quickly output a family of simple, light, stiff and buckling-resistant layouts.